Assumption is that std dev of distribution grows as square root of time (or variance scales linearly with time). Quotes implied vol is in annual terms. Easy back of envelope calculation: 1-day range ~= 1-yr range / sqrt(num of trading days in a year).

It's interesting how confused everyone is about the price distribution as opposed to the returns distribution(admittedly, at first glance I made this mistake too). No one has even given you a correct answer yet. Under Black-Scholes assumptions the price distribution is certainly not symmetric, it has a positive skew.

The truth is that your question is not even well posed. You need to specify whether you mean standard deviation with respect to real world probabilities or risk neutral probabilities because the standard deviation of a lognormal random variable depends on the mean of the underlying normal random variable.

So in fact, this question is more interesting than it seems at first glance and should be given more attention.